$f(x,y) = \dfrac{x^{3}}{3} + y^{2}$ What is $\dfrac{\partial f}{\partial x}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $x^{2}$ (Choice B) B $x^{2} + 2y$ (Choice C) C $x^{2} + y^{2}$ (Choice D) D $2y$
Solution: We want to find $\dfrac{\partial f}{\partial x}$, which is the partial derivative of $f$ with respect to $x$. When we take a partial derivative with respect to $x$, we treat $y$ as if it were a constant. Let's break $f(x, y)$ down term by term. $\begin{aligned} &\dfrac{\partial}{\partial x} \left[ \dfrac{x^3}{3} \right] = x^2 \\ \\ &\dfrac{\partial}{\partial x} \left[ y^2 \right] = 0 \end{aligned}$ Adding the terms back together, we get the partial derivative. In conclusion: $\dfrac{\partial f}{\partial x} = x^2 + 0 = x^2$